Convergence of Spectral-Grid Method for Burgers Equation with Initial-Boundary Conditions
Annotatsiya
In this study, initial-boundary value problem for the Burgers equation is solved using the theoretical substantiation of the spectral-grid method. Using the theory of Green's functions, an operator equation of the second kind is obtained with the corresponding initial-boundary conditions for a continuous problem. To approximately solve the differential problem, the spectral grid method is used, i.e. a grid is introduced on the integration interval, and approximate solutions of the differential problem on each of the grid elements are presented as a finite series in Chebyshev polynomials of the first kind. At the internal nodes of the grid, the requirement for the continuity of the approximate solution and its first derivative is imposed. The corresponding boundary conditions are satisfied at the boundary nodes. A discrete analogue of the operator equation of the second kind is obtained using the spectral-grid method. The convergence theorems for the spectral-grid method are proven and estimates for the method's convergence rate are obtained. To discretize the Burgers equation in time on the interval [0,T], a grid with a uniform step of <img src=image/13435533_01.gif> is introduced, i.e. <img src=image/13435533_02.gif>, where <img src=image/13435533_03.gif> - given number. Numerical calculations have been carried out at sufficiently low values of viscosity, which cannot be obtained by other numerical methods. The high accuracy and efficiency of the spectral-grid method used in solving the initial-boundary value problem for the Burgers equation is shown.